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Upgrading procedure (Posted on 2015-11-17) Difficulty: 3 of 5
The equation a3+b3=c4
has infinite number of positive integer solutions.

Find the two lowest values of S (S=a+b+c).

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution Comment 1 of 1
Strictly speaking the below list is in order by sum of a+b rather than a+b+c, as evidenced by a+b+c=2016 preceding a+b+c=2010, to take the first such example. However, it does include all a+b under 10,000 and thus all a+b+c under 10,000. Only one a+b+c listed is above that value and thus there might exist unlisted entries above 10,000 and below 10,290, the only a+b+c listed that's over 10,000.

The two lowest values sought are 6 and 36.

a b c      a+b+c
2 2 2      6
9 18 9      36
32 32 16      80
28 84 28      140
70 105 35      210
162 162 54      378
65 260 65      390
144 288 72      504
273 364 91      728
126 630 126      882
266 665 133      1064
512 512 128      1152
469 603 134      1206
456 760 152      1368
455 1001 182      1638
217 1302 217      1736
793 854 183      1830
756 945 189      1890
448 1344 224      2016
737 1072 201      2010
730 1241 219      2190
729 1458 243      2430
417 1807 278      2502
728 1729 273      2730
1250 1250 250      2750
344 2408 344      3096
721 2060 309      3090
1120 1680 280      3080
702 2457 351      3510
211 3165 422      3798
665 2926 399      3990
1110 2590 370      4070
1705 2046 341      4092
604 3473 453      4530
1628 2849 407      4884
513 4104 513      5130
1136 3905 497      5538
2592 2592 432      5616
1040 4160 520      5720
386 4825 579      5790
2340 3276 468      6084
217 5642 651      6510
1617 4312 539      6468
2304 4608 576      7488
3354 3913 559      7826
730 6570 730      8030
1474 6633 737      8844
3185 5096 637      8918
3666 5170 658      9494
2268 6804 756      9828
4802 4802 686      10290

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)

 For tot = 2 To 10000
   For a = 1 To tot / 2
      DoEvents
       b = tot - a
       c4 = a * a * a + b * b * b
       c = Int(Sqr(Sqr(c4)) + 0.5)
       If c * c * c * c = c4 Then
         Text1.Text = Text1.Text & a & Str(b) & Str(c) & "      " & a + b + c & crlf
       End If
   Next a
 Next tot


 Text1.Text = Text1.Text & crlf & " done"
  
End Sub


  Posted by Charlie on 2015-11-17 18:57:42
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